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Separating doubly nonnegative and completely positive matrices

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Abstract

The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NP-Hard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then to separate a given DNN but non-CP matrix from the cone of CP matrices. We describe two different constructions for such a separation that apply to 5 × 5 matrices that are DNN but non-CP. We also describe a generalization that applies to larger DNN but non-CP matrices having block structure. Computational results illustrate the applicability of these separation procedures to generate improved bounds on difficult problems.

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Authors and Affiliations

  1. Department of Applied Mathematics and Computational Sciences, University of Iowa, Iowa City, IA, 52242, USA

    Hongbo Dong

  2. Department of Management Sciences, University of Iowa, Iowa City, IA, 52242, USA

    Kurt Anstreicher

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  1. Hongbo Dong
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  2. Kurt Anstreicher
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Correspondence to Kurt Anstreicher.

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Dong, H., Anstreicher, K. Separating doubly nonnegative and completely positive matrices. Math. Program. 137, 131–153 (2013). https://doi.org/10.1007/s10107-011-0485-8

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  • DOI: https://doi.org/10.1007/s10107-011-0485-8

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